The solutions of the quaternion matrix equation AXε + BXδ=0

In this paper, we discuss the quaternion matrix equation AX(epsilon) + BX delta = 0, where epsilon is an element of {I, C}, delta is an element of {dagger,*} and I, C, dagger, * denote the identity, involutive automorphism, involutive anti-automorphism and transpose, involutive automorphism and anti-automorphism and transpose, respectively. Firstly, we transform the given matrix equation into the matrix equation (A) over tildeY(epsilon) + (B) over tildeY(delta) = 0 with complex coefficient matrices and quaternion target matrix Y by utilizing the regularity of (A, B), where (A) over tilde = PAQ and (B) over tilde = PBQ are two complex matrices with P, Q being two invertible quaternion matrices. Secondly, we show that the solution can be obtained in terms of Q, the Kronecker canonical form of ((A) over tilde, (B) over tilde) and one of the two invertible quaternion matrices which transform ((A) over tilde, (B) over tilde) into its Kronecker canonical form. Meanwhile, we also decouple the transformed equation into some systems of small-scale equations in terms of Kronecker canonical form of ((A) over tilde, (B) over tilde). Thirdly, we determine the dimension of the solution space of the equation in terms of the sizes of the blocks arising in the Kronecker canonical form. Finally, we give the necessary and sufficient condition for the existence of the unique solution. (c) 2023 Elsevier Inc. All rights reserved.